Scilab Function
Last update : 6/10/2005

intg - definite integral

### Calling Sequence

  `[v,err]=intg(a,b,f [,ea [,er])`

### Parameters

- `a,b`: real numbers
- `f`: external (function or list or string)
- `ea, er`: real numbers
- `ea`: absolute error required on the result. Default value: 1.d-14
- `er`: relative error required on the result. Default value: 1.d-8
- `err`: estimated absolute error on the result.

### Description

`intg(a,b,f)` evaluates the definite integral from `a` to `b` of `f(t)dt`. The function `f(t)` should be continuous.

The evaluation hopefully satisfies following claim for accuracy: `abs(I-v)<= max(ea,er*abs(I))` where I stands for the exact value of the integral.

`f` is an external :

If `f` is function its definition must be as follows `y = f(t)`

If `f` is a list the list must be as follows: `list(f,x1,x2,...)` where `f` is a function with calling sequence `f(t,x1,x2,...)`.

If `f` is a string it refers to a the name of a Fortran function or a C prodedure with a given calling sequence:

In the fortran case the calling sequence should be `double precision function f(x)` where `x` is also a double precision number.

In the C case the calling sequence should be `double f(double *x)`.

### Examples

    //Scilab function case
    function y=f(x),y=x*sin(30*x)/sqrt(1-((x/(2*%pi))^2)),endfunction
    exact=-2.5432596188;
    I=intg(0,2*%pi,f)
    abs(exact-I)

    //Scilab function case with parameter
    function y=f1(x,w),y=x*sin(w*x)/sqrt(1-((x/(2*%pi))^2)),endfunction
    I=intg(0,2*%pi,list(f1,30))
    abs(exact-I)

    // Fortran code case (Fortran compiler required)
    // write down the fortran code
    F=['      double precision function ffun(x)'
       '      double precision x,pi'
       '      pi=3.14159265358979312d+0'
       '      ffun=x*sin(30.0d+0*x)/sqrt(1.0d+0-(x/(2.0d+0*pi))**2)'
       '      return'
       '      end'];
    mputl(F,TMPDIR+'/ffun.f')
    // compile the fortran code
    l=ilib_for_link('ffun','ffun.o',[],'f',TMPDIR+'/Makefile');
    // incremental linking
    link(l,'ffun','f')
    //integrate the function
    I=intg(0,2*%pi,'ffun')
    abs(exact-I)

    // C code case (C compiler required)
    // write down the C code
    C=['#include <math.h>'
       'double cfun(double *x)'
       '{'
       '  double y,pi=3.14159265358979312;'
       '  y=*x/(2.0e0*pi);'
       '  return *x*sin(30.0e0**x)/sqrt(1.0e0-y*y);'
       '}'];
    mputl(C,TMPDIR+'/cfun.c')
    // compile the C code
    l=ilib_for_link('cfun','cfun.o',[],'c',TMPDIR+'/Makefile');
    // incremental linking
    link(l,'cfun','c')
    //integrate the function
    I=intg(0,2*%pi,'cfun')
    abs(exact-I)

### See Also

`intc`, `intl`, `inttrap`, `intsplin`, `ode`,

### Used Function

The associated routines can be found in routines/integ directory :

dqag0.f and dqags.f from quadpack